
From this we generate a minimising sequence \((\mathbf {q}^n, u^n, \mathbf {w}^n)_{n\ge 0}\)  admissible to (REF ). The rest of the proof
is standard, see e.g. [1]}. We show the constraint equation
is continuous with respect to the weak topology on \(X\triangleq H^1([0,1],\,\mathbb {R}^{d\times M})\times L^2([0,1],\,V)\times L^2([0,1],\,\mathbb {R}^{d\times M})\)  i.e. \(e(\mathbf {q}_t^n,\,\mathbf {w}_t^n,\, u_t^n)\rightharpoonup e(\mathbf {q}_t,\,\mathbf {w}_t,\, u_t)\)  where \(e(q,\,w,\,u) \triangleq \dot{q} - u(q) -\sqrt{\nu (q)}w\) . Then,
\(\langle \sqrt{\nu (\mathbf {q}_t)}\mathbf {w}_t - \sqrt{\nu (\mathbf {q}_t^n)}\mathbf {w}_t^n,\,\phi \rangle \lesssim \nu _\text{inf}\langle \mathbf {w}_t - \mathbf {w}_t^n,\,\phi \rangle \rightarrow 0\,,\quad \forall \phi \in L^2([0,1],\,\mathbb {R}^{d\times M})\,.\) 
